The Glasgow giants struck first through El-Hadji Diouf, who scored his first Ibrox goal, in the 38th minute when he received a pass from Steven Whittaker before firing past Killie keeper Anssi Jaakkola from 10 yards.
It looked destined for a draw until Clancy was the unfortunate Killie player who knocked the ball into the back of his own net when Vladimir Weiss put in a dangerous cross in the 87th minute.
Rangers boss Walter Smith felt his side was hardly done by with the penalty awarded to Kilmarnock as ‘Gers striker Nikica Jelavic was fouled in similar fashion but received nothing from referee Alan Muir.
“Were Kilmarnock not fortunate? Was Jelavic's not a penalty? He turned the boy and was pulled down in the box and it was a penalty kick,” he said.
"If Kilmarnock's penalty is a penalty and that one isn't, then I've wasted my time in football for a long while. I couldn't understand the decision.
"Also, I don't mind any of the opposition wasting time at any given time, I don't mind that all. But, for the Kilmarnock goalkeeper to go through 90 minutes wasting the amount of time that he wasted and for the referee not to speak to him once, was an amazing factor.
"It was absolutely amazing. In the European games, if our goalies try to waste time, the referees warn them early. Here, they don't.
"It's a fairly amazing one so I don't think anyone could say anyone got away with any decisions. In fact, quite a few of the decisions over the 90 minutes were questionable."